An Optimal Poincark Inequality for Convex Domains of Non-negative Curvature

I. CHAVEL & E. A. FELDMAN

Communicated by J. SERR1N

1. Introduction

In this note we estimate from below the first non-zero eigenvalue of the Neu- mann problem of two dimensional convex domains of non-negative Gaussian curvature.

The Neumann problem is described as follows: Let M be an oriented manifold with fixed complete Riemannian metric, and let A be the Laplace-Beltrami operator associated with this metric, acting on realvalued functions on M. Also let f2 denote a fixed compact connected submanifold of M whose boundary c?f2 is continuous and piecewise of class C ~, and whose interior f2 ~ has the same dimension as M. Our task is to study the eigenvalues problem

Au+2u=O

on I2 subject to the boundary condition

~u - - = 0 dn

on 6f2, where ~/~u is the normal derivative of u at points of ~2 which are of class C;. Certainly 2 = 0 is an e igenvalue- i ts eigenfunctions are the constant functions. All other eigenvalues are known to be positive, and the lowest positive eigenvalue 21 is characterized by the minimum principle [3]

Hgrad ull 2 dA

21 = min ~ u ~ ~ u2 d A '

D

where dA is the induced Riemannian measure on M, a f is the class of C < real- valued functions on ~ for which

~. udA=O, f2

and II I[ is the length of tangent vectors to M determined by the Riemannian metric.

We now state our main result.

264 I. CHAVEL ~,Z E. A. FELDMAN

Theorem. Let f2 be a geodesically convex two-dimensional compact domain in M with continuous piecewise C ~ boundary, and assume that the Gaussian curvature of M is non-negative and bounded. Then

;~1 > ~2/d2

where d denotes the diameter of (2.

The inequality is best possible in the sense that for rectangles in the plane with sides of length a, b, where a is kept fixed, we have 21 d 2 -~ ~ as b ~ 0. We also note that the above theorem extends a result of L. E. PAYNE 8,: H. F. WEINBERGER [7], where the conclusion is proved for convex domains in the plane. Our proof ex- tends theirs as well, in that it relies heavily on their method of reducing the dis- cussion to a one-dimensional problem. We remark that their original argument has a small gap (cf. the first paragraph of [7, p. 290]), which we correct in Lemma 2.2.

In our proof we make use of several fairly elementary properties of geodesically convex regions. The reader can check the truth of these statements himself or refer to w 4 of [5], where proofs can be found. We also use [4] and [8] as standard references for the needed differential-geometric information.

2. Three Lemmata

The following lemma is the main comparison result that we need. We refer the reader to [7, p. 288] for a proof.

Lemma 2.1. Let p: [0, L] -~ [0, ~ ) be a continuous piecewise C 2 function such that p(y)>O for y~(O, L). I f p"(y)<O at all points of (0, L) for which pc C 2, then for every continuous piecewise C 1 Ji~nction u: [0, L] --. R satisfying

L

Su(y) p(y)dy=O 0

we have g L

I p (y) (u'(y))2 dy > ( 2/L p 0,) (u (y))2 dy. 0 0

Our second lemma corrects the above-mentioned gap in [7]. In this lemma and throughout the rest of the paper S(M) will denote the unit tangent bundle of M and re: S ( M ) - , M the standard projection map. For q e M , Sq(M) will be the fiber over q in S(M), viz. rc-l(q), and for any set E in M we will often write S(M)]E in place of n-l(E). Also, for any domain D in M we will let A(D) denote the area of D. M will henceforth be two-dimensional.

Lemma 2.2. Let ~2 be a geodesically convex compact domain in M with con- tinuous piecewise C ~ boundary. Let u: M ~ I R be continuous and such that the integral of u over f2 is O. Then there exists a geodesic { which divides f2 into two regions f21, 0 2 of equal area such that the integral of u over f21 and the integral of u over s 2 are both O.

Proof. Let co: [0, L] ~ M be the boundary curve of f2, parametrized with respect to arc length. Then co is continuous and piecewise of class C 2. Let exp

An Optimal Poincar6 Inequality 265

denote the exponential map of the tangent bundle of M to M, and for vr set

s(v)=sup{so:expsv~(2 for sE[0, s0)}, and

/ , , (s)=expsv for s~[O,s(v)].

Thus if s (v) = 0 then the image of r is precisely the point zr(v), and if s (v) > 0 then {,: is a geodesic segment which divides s into two convex sets ~2i, I,. ~ and ~2RW ~ which lie to the left and right of f,., respectively.

Now let I o = {v~S(M)IO~2: s(v) > 0}

and note that Io is an open subset of S(M)10Q. Then the function that maps v~lo to A(~2R,,~) defines a cont inuous map r: Io ~ (0, ~) .

Set A=A(g2). We want to construct a cont inuous map t ~ v(t)~S~m(M)c~ Io (i.e., a cross-section of lo) such that r(v(t))=A/2 for all t6 [0 , L].

Case I. Let t be a regular point of co, ~ (t) the velocity vector of co at co (t), and v(t, O)~S,om(M) the vector which makes the positive angle 0 with ~(t). Assume that s lies to the left of co. Then O~r(v(t, 0)) is cont inuous and mono tone in- creasing on (0, ~), and satisfies

lim r(v(t, 0)) = A, ~imr(v(t, 0)) = 0 . 0~r~

Thus there exists a unique value 0,~(0, Tr) such that r(v(t,O))=A/2. Set v (t) = v (t, 0,), and assume to is a regular point of co. We show that t ~ v (t) is con- t inuous at t o. Let 0,o~(0, 7r) as above. Then we can pick ~>0 so small that

from which one concludes

[0,o - e, 0,o + e] = (0, r0,

o < r(v (to, 0,o- < A/2,

A/2 < r(v(to, O,o + e)) < A .

By the continuity of r(v) near V(to), we can find a f i>0 such that if It-tol < 6 then co(t) is a regular point of co, and

0<r (v ( t , O,o- e)) < A/2,

A/2<r(v(t, 0,o +~:)) < A . Thus

Oto--C ~Ot ~Oto ~t-C.

for I t - t o I<& and this case is disposed of.

C a s e II. If s is a singular point (i.e., a corner) of co, and if we let ~_(s), ~+(s) denote the respective left and right velocity vectors of co at s, then the convexity of ~ implies that the positive angle O0 from ~_(s) to ~+(s) is less than ~. Let 0 measure the positive angle from ~_(s) and let v(s,O)~&,~(M) be as in Case I. Then O~r(v(s,O)) is cont inuous and mono tone increasing for 0 o < 0 < ~ , and satisfies

lira r(v(s, 0))= A, lira r(v(s, 0) )=0 . 0 ~ 0400

266 I. CHAVEL & E. A. FELDMAN

~'{t)

Figure 1.

(s)

Thus there exists a unique 0~(0, 7r) such that r(v(s, O~))=A/2. Set v(s)=v(s, Os), and note that v(s)elo. The map t - . v ( t ) is continuous from the left at s as in Case I.

We now show that v(t) is continuous at s from the right. Notice that

Io = {v(t, 0): t is regular, 0 < 0 < ~ } w {v(t, 0): t is singular, Oo(t)< t<lr}

and that r: I0 -oR extends continuously to ]o (we also denote its extension by r). Let t, be a sequence of regular points such that t, > s for all n, and t, -o s as n --* oo. We show that lira v(t,)=v(s). Indeed let v0 be a cluster point of {v(t,)}; certainly one exists by the compactness of S(M)I~f2. It suffices to show Vo=V(S). Now Vo e io and r(vo)= lira r (v (t,))= A/2. But v (s) is the unique element of io c~ SmI~(M), with this property.

We can now prove the lemma. The geodesic segment {,,co) meets ~f2 exactly once more, say at ~o (T), where Te (0, L). We note that v (T) = - d',,(o)(co (T)). Define

qo(t)= ~ u d A - S u dA. ~L(v (t)) OR (v ( t ) )

Then ~o is continuous on [0,1r] and ~o(0)=-(p(T). Hence there exists some to e [0, T] such that ~o (to)= 0. The lemma is proven.

Our final lemma will be an area comparison theorem.

Lemma 2.3. Let D be a convex compact domain in M and 7 a geodesic in D realizing the diameter of D. Also let w denote the maximum length of all geodesic segments in D which are perpendicular to 7 (the "width" of D). We assume that K(p) is non-negative and bounded for all p6M, and set

tr = sup {K(p): p 6 M } < ~ ,

[rain(w/Z, 7r/(2 l/F)) for ~c> 0 e , /

~w/2 Jor K =0.

Then A(D), the area olD, satisfies

A (D) > 1/3 c2/4.

Proof. Let pc?, be the point at which the geodesic segment in D orthogonal to 7 has length w. Then 2c<_w<_ diameter of D. We shall assume that both 7 and a, the geodesic through p orthogonal to 7, are parametrized with respect to arc length in such a way that

p=7(0 )=a (0 ) and ?([0, c ] ) ~ D , a([O,c])~D.

An Optimal Poincar6 Inequality 267

We next note that by the arguments of [2], [6] the disk of radius rc/l,/K about the origin in the tangent space to M at p is mapped diffeomorphically (via the exponential map) onto its image in M. Such a metric disk in M about p, of radius p, will be denoted by Bo(p); then the standard Sturmian arguments also show that the geodesic curvature of the boundary of B o (p)is strictly_positive for pc(0, 7r/(2]/K)). We conclude that B o (p) is convex for all p ~ (0, re/(2 l/N)) [4, p. 160].

Now let q be any number in (0, c) and set q =7 (cl), r = a (Cl). Then d(q, r)< ~/1/tc by the triangle inequality, and the unique geodesic joining q and r is completely contained in B~/~2V~(p). Let S 2 (~c) be the standard 2-sphere of constant curvature K with distance function d~(, ); and let p, q,_reS2(~c) be such that d~(p,q)= d~(p,_r)=c 1 and such that the minimizing geodesic segments p q, pr_ intersect orthogonally at p. Then by the Rauch comparison theorem [4, p. 180]

c 1 <d~(q_,s r).

If we apply the Rauch comparison theorem again and compare with Euclidean space we finally get

q <d(q, r)<=l/2 c 1.

We now finish the argument. Let IR 2 be the Euclidean plane with distance function do(, ) and let ~,g/,?elR 2 be such that d(p, q)=do~,F1), d(q, r) = do (~/, ?), and d(r, p) = do (?, .o). Let T be the geodesic triangle in B~/~2g~(p) determined by p, q, r and 7 ~ the triangle in IR 2 with vertices p, q, r. A(T) is the area of T and Ao(T ) will denote the Euclidean area of T. Then A. D. ALEXANDROV'S area com- parison theorem [1, p. 399] gives

A(D)>=A(T)>=Ao(T), and also

Ao(T)>t/-3 ef/4 because

Cl _-< do(~/,~)_-< l/2 c1. Thus

A (D) >= ]/3 c 2/4

for all Cle(0, c), which implies our result.

Corollary 2.4. Let D, 7, and w be as in Lemma 2.3./fb is any number in (0, re/(21/~)) and

(1) (A(D))~ ~ (3a-/4) 6,

then w<__&

thus "'small area implies small width". Furthermore let ~ denote the rotation of any tangent space to M by ~/2 radians,

and 7' the velocity vector oJ'7. Construct Fermi coordinates along 7 by setting

q0 (x, y) = exp x ( - z y'(y))

where 7 is defined on [a, b]. I f A(D) satisfies (1) J0r be(0, 7r/(2 l/~) ), then

o = ~o ( ( - 6, 6) • [ . , hi) .

268 I. CttAVEL & E. A. FELDMAN

3. The Convexity of Small Domains Expressed Analytically

In this section we shall consider the convex compact domain D of Lemma 3.2, with the geodesic segment 7: [a, b] ~ M realizing its diameter. Also we shall assume that 6e(0, rc/(Zv/~)), that the area of D satisfies (1), and that we have Fermi coordinates based on 7 in a neighborhood of M containing D, as in Corollary 2.4.

Consider the part of the boundary of D in the "right half-plane" q~([0, 6] x [-a, b]). At first it may start along the geodesic y=a, but it must then leave the axis and makes its way in a continuous piecewise C ~ manner to the geodesic y=b without ever possessing a horizontal tangent (except possibly at y=a or y = b). Therefore

Figure 2.

this part of ~D can be represented as the graph of a continuous piecewise C ~ func- tion x+ (y) such that (i) x+ (y)>0 if ye(a, b) and (ii) all left- and right-hand deriva- tives of x(y) on (a, b) are bounded.

In a similar fashion the part of 0D in the "~left half-plane" corresponds to a function x_ (y) < 0 on (a, b). Put

(y) = x + (y) - x _ (y).

Then )2 (y) will be positive, continuous and piecewise C ~' on (a, b), with bounded left- and right-handed derivatives. We will show that at the points yoe(a,b) where ))(y) is regular there holds

(2) ~ " ( y o ) < 0 .

To establish (2) we consider x+ (y) and - x (y) separately. In fact it is enough to treat only x+(y), since the argument for -x_(y) is almost the same.

We now note that on q~((-6, 6)x [a, a] ) the tangent vector fields #~p/~x and 0~p/#y are always orthogonal, and #~p/#x always has unit length. If we set

G(x, y)= IIc~/ayl[ 2 (x, y),

An Optimal Poincare Inequality 269

then

(3)

and

G(0,y)= 1, G~(0, y)=0

(6~)x~ (x, y) + K(~p (x, y)) 6~ (x, y) = 0

(6)

By TAYLOR'S formula

(the subscr~ts x ,y . . , denote the indicated partial differentiation). Since 6~(0, 7z/(2 ]/~c)) we have by usual Sturmian arguments

(4) Gx(x, y)NO, O<G(x, y)N1

on ( - 6, 6) x [a, b]. For the rest of this section we will write x (y) for x+ (y) and put

co (y) -- ~o (x (y), y), a < y < b.

The geodesic curvature % of co at co(y) is given by the relation [8, p. 128]

G- ~ (x(Y), y)Ild(y)l[ 3 ~g(co(y))

(5) = G x ( x ( y ) , y ) { 1 4 (x'(y)) 2 } x'(y)Gy(x(y),y) G(x(y),y) q 2G(x(y),y) x"(y),

at least for points y~(a,b) about which x(y)~C 2. Let Yo be fixed in (a,b) and assume that x (y)~ C ~ in a neighborhood of Yo. We intend to show that the assump- tion x"(y0)>0 leads to a contradiction.

Set c~= x'(yo). In the ensuing calculations G, Gx, Gy . . . . will denote the values of the indicated functions at (X(yo), Yo). Furthermore, we let z(y) be the C a func- tion (a, b) such that the path

(y) = ~o (z (y), y)

has zero geodesic curvature and satisfies

a(yo)-= co(yo), o"(yo)-- co'(yo).

Of course u=z'(yo). Letting fl-~z"(yo) , we have also

G f l cd] uGy

t 2/3 t 3 c( t ) z (Y~176 6 '

where c(t) is bounded. Finally for any given sufficiently small s, set

/ s 13 s z c(s) \ 3(t, s )=x(yo)+t i ~ + 2 - + ~ C - )

v'(t, s)-~ q~(3 (t, s), yo + t) $

L(s)-- ~ { [I a' (t)[I - 11(6 v/a t) (t, s)II} dr. 0

270 i. CHAVEL & E. A. FELDMAN

Thus L(s) is the difference of the length of the geodesic o from o(yo) to o(yo+s) and the length of the image of a straight line in ( - b , g ) x [a, b] from O(yo) to a (Yo + s). For sufficiently small s it is clear that L(s) < O, i.e., s = 0 is a local maximum of L(s). By direct calculation we shall show below that E"(0)> 0 and that/2 (0)= /2' (0)= 0. Therefore for small positive s we have L(s)> 0, which is a contradiction.

That 0 = L(O) =/2 (0) = E' (0)

we leave to the reader to verify. We next note that

d 2 ~V

A rather long and tedious calculation shows that then

{1+G~ fGx G~~ ~ G/JL + + c-,-, ~b.. | 2 ~-G[Je+3Gy~ / 2 " ( 0 ) -

- 4 2Gfl2+~Gy[3/3+G~

- the last line follows from the preceding one by (6). By the convexW of D we have ~Cg (CO (y)) > 0, which implies by (5) that

Then x" (Yo) > 0 implies

0<f l , O<ctGy,

which in turn shows that/2"(0) > 0. Our claim is proven.

4. Proof of the Theorem

Step I. We first consider the set VcIR x (S(M)If2) given by

V={(y,v):expsve~2 for s~[O, y]}.

Let Pl : IR x S(M) --* ~, and P2: IR • S(M) ~ S(M) denote the respective projec- tions. Then for veS(M)lf2 the set {pa(p2) -l(v)} is a closed interval (for a given pe~3(2 it could however possibly consist of {0} alone), from which one sees that V is compact.

Consider the map ~: V--* S(f2) defined by

y

An Optimal Poincar6 Inequality 271

Then for each (y, v)e V, { - ~ ~(y, v), r v)} is a positively oriented or thonormal frame at ~(~_ (y, v)). We now set

v (x, y, v) = exp x ( - z ~ (y, v)) and

G(x, y, v)= II(0 v/~y)(x, y, v)[I 2

viz., for each vep2(V ) the function v yields the Fermi coordinate system of w based on the geodesic determined by v. The various properties of G developed in w 2 of course continue to hold.

Step II. Assume that f: C2 ~ IR is the desired eigenfunction, that is

~" Ilgrad f [I 2 dA /~1~--- D

~f2 dA s

Set rn = sup {If(p)], Ilgrad f(p)l[ }.

p~D

If e is any given value in (0,~), then by (3), (4) and the compactness of V there exists a 61E(0, 1] c~ (0, ~/(21/~)) such that for [xl < 61 and (y, v)e V we have

(7) 1 - e<G(x , y, v)< 1 <G -1 (x, y, v)< 1 +e,

and also such that for F(x,y, v)=f(v(x, y, v)), Ixl <61, and (y, v)e V we have

(8) IF 2 (x, y, v) - F z (0, y, v)l <

(9) I(Fy) 2 (x, y, v) - ( f , ) 2 (0, y, v)[ < e.

We now apply Lemmata 2.2 and 2.3. Divide f~ into a finite number of convex regions ~s of equal area (each with diameter d s and "wid th" less than 61) such that

(10) ~ fdA = 0 Dj

for every j. We fix one such Qs" Let 7 / [ 0 , dj]--* f2 s be the geodesic which realizes the diameter of f2j, and pick vjeS(M)[?fJj so that 7j is given by the formula Vs(y)=exp y vj. For the rest of the discussion we shall write F(x, y) for F(x, y, v~) and G (x, y) for G (x, y, v j).

Step III. We note that

and set dA]Oj= G~(x, y) dx dy

Aj = A (f2s). We therefore have

[I F2(x,Y) d A - ~ F2(O,y)dxdyl f2j Dj

~ {IF 2 (x, y) -- F 2 (0, Y) I+ F2 (0, y) (1 - G - ~' (x, y)[} dA ~j

<(1 + m 2) Aj

by (7) and (8).

272 I. CtlAVEL & E.A. FELDMAN

Now let ~ (y) be the width of ~2j at 7j(Y), namely, the length of the maximal geodesic segment through 7j(Y) which is perpendicular to ~j(y) and is contained in Oj. Then ~ (y) is continuous and piecewise C a on [0, dj], positive on (0, d j), and satisfies ~" (y) < 0 by w 3.

Now note that [[ grad f [[ 2 = (Fx)2+ (G) - 1 ( 5 ) 2 .

Then by (4) and (9)

[S (Fy)2(x, Y) G- I (x, Y) dA - ~ (Fy) 2 (O, y) dx d y[ .Qj g)j

=< ~ {1(5) 2 {x, y ) - (Fy) 2 (0, y)] G-l(x, y)+ (Fy)2 (0, y)]G-~(x, y ) - 11} dA ~j

=<2(m 2 + l) Aj e,

which implies Hgrad f[[ 2 dA>__ ~ (Fy) 2 G -1 dA

~j ~2j

>= ~ (~)~ (o, y) dx d y - 2 ( m 2 + 1) A~ ~j

dj > S (Fy) 2 (0, y).~(y) dy - 2 ( m 2 + 1) Aj e.

0

But by Lemma 2.1 and the fact that ~"(y)__<0, we get

dj (5) 2 (o, y) ~ (y) dy

0

~2 ( dj dj \ - 1 dj 2~ > - ~ ~ F2(O,y)Yc(y)dy- ( ~ s (! F(O, y) s =dr (o o

1.C 2 > ~ - { ~ F 2 (x, y) dA - ( A j ) - l ( ~ V(O, y) d x dy) 2 - ( 1 + m 2) Aj ~}.

~j ~j

Finally (10) implies

] ~ F(O, y)dx dy] = ] ~ F(x, y) dA - ~ F(O, y) dx dy]

=< ~ {[F(x, y ) -F (0 , y)[ + IF(0, y)[ ]l - G-�89 y)[} dA 9j

< ( m + l ) A j e .

We therefore have

21 ~f2 dA = ~ ]lgrad/]l 2 dA

~2 >= d 2 { ~ f2 dA_{(m 2 + l ) + ( m + l ) 2 } A j e } - 2 ( m Z + l ) A j e

~j

- d 2 ~ f 2 d A - ~-[(m2+l)+(m+l)2]+2(m2+l) Aje. ~j

An Optimal Poincare Inequality 273

Let k denote the expression in braces in the last line. If we sum the above in- equality over j we obtain

21 ~ f 2 dA >(Tz2/d2) ~ f 2 d a - k A ( Q ) ~. i2 f2

Since e was arbitrary the theorem now follows.

Note: This research was partially supported by N.S.F. Grant No. 27962.

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Graduate School & University Center The City University of New York

(Received April 15, 1976)